Tilings from Graph Directed Iterated Function Systems

TL;DR

This paper introduces a novel method for creating self-referential tilings of Euclidean space using graph directed iterated function systems, emphasizing a combinatorial structure called a pre-tree, and characterizes when such tilings are congruent.

Contribution

It presents a new construction technique for self-similar, quasiperiodic tilings from graph directed iterated function systems, with a focus on balanced tilings and their congruence conditions.

Findings

Balanced tilings have a finite set of prototiles.

Balanced tilings are quasiperiodic but not periodic.

A condition for congruence of balanced tilings is established.

Abstract

A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system, based on a combinatorial structure we call a pre-tree, is introduced. In the special case that we refer to as balanced, the resulting tilings have a finite set of prototiles, are quasiperiodic but not periodic, and are self-similar. A necessary and sufficient condition for two balanced tilings to be congruent is provided.